If at each boundary point $f(z)$ and $g(z)$ are nonzero and $g(z)$ is not a negative real multiple of $f(z)$, then $f$ and $g$ have the same number of zeros inside.Suggest a slogan
Let \(D=\theset{z : \abs{z-z_0}<R}\), and let \(f,g\) be holomorphic on an open set containing \(\overline D\). If \[ \abs{f(z) - g(z)} < \abs{f(z)} + \abs{g(z)} \quad\text{for all } z\in\bd D, \] then \(f\) and \(g\) have no zeros on \(\bd D\) and have the same number of zeros in \(D\), counted with multiplicity.
State Rouche's theorem in its symmetric form.